Polynomials: Definition and Operations

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Algebraic Geometry: Rings and Bodies

Explores algebraic geometry, focusing on rings, bodies, quotient rings, and irreducible polynomials.

Polynomials: Theory and Applications

Covers the theory of polynomials, including definitions, properties, and applications.

Limits in Function Analysis

Introduces the concept of limits in function analysis and demonstrates their application through various examples.

Vector Spaces: Properties and Examples

Explores vector spaces, focusing on properties, examples, and subspaces within a practical exercise on polynomials.

System Equivalence

Explores system equivalence, state-space representation, transfer functions, and Euclidean rings, emphasizing unimodular matrices and their properties.

Linear Combinations and Vector Spaces

Introduces linear combinations in vector spaces, operations, and polynomials of degree 2.

Differentiable Functions and Lagrange Multipliers

Covers differentiable functions, extreme points, and the Lagrange multiplier method for optimization.

Advanced Analysis I: Taylor with Integral Remainder

Covers Taylor series with integral remainder for open intervals and their applications in mathematical analysis.

Euclidean Division: Uniqueness and Remainder

Explores Euclidean division for polynomials, emphasizing uniqueness of quotient and remainder.

Ideals: Polynomials and Definitions

Explores ideals in K[X], including PGCD, uniqueness, coprimality, and theorems of Bézout and Gauss.